However, the length of this exploration phase increases under delayed feedback, as the reward associated with each action returns at some unknown time in the future. Since no updates occur in this phase, the learner cannot utilise information from the rounds with observed rewards. Ideally, we would like to leverage new information as soon as it becomes available. Thus, we introduce a penalisation term into the objective function, an idea that we borrow from the linear and contextual linear bandits, where one can derive a closed-form maximum likelihood estimator [1]}, [2]}. In the generalised linear setting, this equates to penalising the log-likelihood. From Equation (REF ) and the conditional independence of the rewards given past actions, one can write the penalised log-likelihood as follows:
\(\mathcal {L}_{t}\left(\theta , \alpha \right) = \sum _{s = 1}^{t}C_{s}^{t}\,\log \left(f\left(Y_{t}\,\vert X_{t}\right)\right)- \frac{\alpha }{2}{\theta }_{2}^{2}\) 
